The Road to Strongly Interacting Bose Gases 

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Two-Particle Problem

This chapter is dedicated to two-particle scattering. We will remind the reader of the ideas behind the system. These ideas are used in Chapter 2 to derive the three-particle scattering. We will discus concepts like the scattering amplitude, scattering cross-section, unitary limit and scattering length, as well as the zero-temperature limit, which allows us to only consider scattering in the lowest collisional channel (s-wave scattering, for bosons). Afterwards, we will summarize the results by introducing a Zero-Range Model, in which case a boundary condition at r = 0 contains all the information about the two-particle scattering. Finally, we will touch upon the subject of Feshbach resonances, which we will use to tune interactions between particles.


Consider two particles of mass m. Their quantum state is described by the Hamiltonian for two particles interacting via a potential U(|r1 −r2|):
p2 p2
H = 1 + 2 + U(|r1 − r2|), (1.1)
2m 2m
where pn is the momentum operator and rn the position operator of particle n.

Center of Mass (CoM) Motion

The first step in solving the problem is to separate the center of mass motion from the relative motion, because we are only interested in the latter. In order to do so, let us introduce the following CoM variables
r1 + r2
rCoM = 2
pCoM = p1 + p2
and also the relative motion variables
r = r1 − r2
µp = pm1 − pm2 .
Here µ=m/2 is the reduced mass of the system of two particles.
Using the CoM and relative variables introduced above the Hamiltonian can be written in the following form,
p2 p2
H = CoM + + U(r)

4m ≡ HCoM + Hrel. (1.2)
Since HCoM commutes with Hrel, we diagonalize them independently. The solutions of HCoM for the free particle are the plane waves (ϕCoM =eikCoM•rCoM with kCoM the wavevector of the CoM system). In the following, we will study the relative motion.

Radial Schrödinger Equation

The Schrödinger Equation for the relative motion can be cast into the following form
2 Δr + U(r) − 2k2 (1.3)
− ψ=0,
2µ 2µ
where Ek = 2k2/(2µ) is the energy of the relative motion for a scattering state (Ek >0).
The Laplacian operator Δr can be rewritten in spherical coordinates using the radius r and angular momentum operator1 L. The Schrödinger Equation becomes
1 L2 2k2 (1.4)
Tr + + U(r) − ψ=0,
2µ r2 2µ
here the operator Tr = − 2 ∂2 + 2 ∂ is the radial kinetic energy operator. The 2µ ∂r2 r ∂r interest of this Schrödinger Equation comes from the separation of the radial and angular part. Since the operator L2 commutes with the Hamiltonian, the radial and angular part of the wavefunction can be separated
ψlm = Rl(r)Ylm(θ,φ)
Rl(r) = 1 ukl(r), (1.5)
where Rl(r) is the radial wavefunction and Ylm(θ,φ) the spherical harmonics. The spherical harmonics are defined by the following differential equation
L2Y m(θ,φ) = 2l(l + 1)Y m(θ,φ), (1.6)
where the quantum numbers l give the eigenvalues of the equation. These spherical harmonics describe the relative angular motion of the two particles.
The quantum number l, indicates in which scattering channel the scattering takes place (this is usually called s-, p-, d-,… wave scattering, for l = 0, l = 1, l = 2,…, respectively). The quantum number m is used to describe the motion in the φ-direction. The possible scattering channels are different for distinguishable and indistinguishable particles. Distinguishable particle will use all channels, indis-tinguishable particles, however, either scattering in even (bosons) or odd (fermions) channels.
When the definition of the spherical harmonics in Equation (1.6) is applied to the Schrödinger Equation we are left with the radial Schrödinger Equation. The introduction of the radial wavefunction ukl(r) in Equation (1.5) allows us to write down the simplified radial Schrödinger Equation
d2 − l(l + 1) − 2µ U(r)+k2 ukl(r) = 0. (1.7)
dr2 r2 2
Note that Rl(r) needs to be regular at the origin and to impose this, the condition ukl(0) = 0 is required. The effective potential Ueff(r) is graphically depicted in Figure 1.2 for different values of l. Figure 1.2.: The van der Waals potentials with a hard-core at r =b and centrifugal con-tributions Ueff(r). The solid black line is the potential for l = 0 and is mono-tonic. The black dashed line is the effective potential with l = 1 and the black dotted line the effective potential with l = 2. The potentials for l = 1 and l = 2 are not monotonic and have a maximum. This maximum creates an effective barrier for particles with an relative energy lower than the barrier. The purple line shows the energy of such a particle pair in the limit of cold collisions k →0. This shows that for sufficiently cold gases, only the l = 0 (s-wave) scattering contributes to the problem.

Scattering Potential

In the previous section, we have introduced the relative potential U(r) without putting constraints on it. Here we will filter out a certain class of potentials.
In the framework of ultracold atomic gases, we are interested in collisions be-tween two neutral atoms. This means that the considered interaction is an induced dipole-dipole interaction and is described by a van der Waals type potential, which is attractive at the long-range and has a hard-core at r =b. For more detailed information about the atomic potentials we refer the reader to the following ref-erences [48, 145–149]. At intermediate-long distances this interaction is described by a −C6/R 6 potential, which has a length scale given by the C 6 parameter: the van der Waals length RvdW = 1 2µC6 .
This length scale has an interesting physical interpretation [150]. At the length scale r ∼RvdW, the potential energy becomes comparable to the collision energy E ∼U(RvdW). This means that the effect of the potential outside of this range vanishes and the wavefunction approximates the free space wavefunction. In other words, the van der Waals length indicates a range over which the potential mod-ifies the behavior of the wavefunction. In the following, we will only discuss the wavefunction outside the range of the interaction and condense the effect of the potential into a boundary condition at r = 0.

Spherical Waves

Outside the range of the potential r ≫RvdW the solutions of the problem are given by a linear combination of an incoming and outgoing wave ϕlm r ≃ 2k2 Ylm(θ, φ) 1 Aoute+i(kr−lπ/2) − Aine−i(kr−lπ/2) , π 2ikr →∞ ≡ Aoutϕlm(out) − Ainϕlm(in). (1.8)
The amplitudes Aout/in are determined by the details of the problem, which we will see in the following section.

Elastic Scattering

In the above, we have written the solutions of the Schrödinger Equation in the basis of the spherical harmonics. The next step is to describe the effect of the scattering potential on each of the spherical waves. We will use the fact that the potential has a finite range RvdW and start by considering an incoming wave outside of the range of the potential r ≫RvdW. Outside the range of the potential, the incoming wave is described by the r ≫RvdW solution. We will consider a virtual propagation of the wave to the scattering center and as soon as the wave starts to feel the potential, it is deformed and finally reflected within the range of the potential. The potential is norm conserving (elastic scattering), so when the wave is coming back out of the range of the potential it will again be the r ≫RvdW solution, but with an acquired phase δl due to the potential U(r) in the r < RvdW region. Let us write this in terms of the wavefunction
2k 2 Ylm(θ,φ) 1 e+i(kr−l π +2δl ) − e−i(kr−l π
ψlm r ≃ ) , (1.9)
2 2
π 2ikr
which at far distance will behave as 1/r sin(kr −lπ/2+δl). If we write this in terms of the incoming and outgoing waves, the wavefunction takes the form ψlm r→∞≃ slϕlm(out) − ϕlm(in), (1.10) where we have introduced sl ≡ Aout/Ain = e2iδl . This sl described the scattering of a wave in channel l due to the potential U(r). This is a trivial case of the scattering matrix diagonal (corresponding to one channel), but as we will see in Chapter 2, it becomes useful when several channels are coupled.
As a final step, let us separate the outgoing wave with no interaction from the part with the phase factor slϕlm(out) = ϕlm(out) + (sl − 1)ϕlm(out), (1.11)
where (sl −1) = 2ie iδl sinδl. When the result is written into the form of the wave-function, it is given by ψlm r →∞≃ ϕlm(out) − ϕlm(in) + (sl − 1)ϕlm(out) r ≃ ϕlm + 2ieiδl sinδl ϕlm(out). (1.12) →∞
This is the result of scattering in a specific channel l. To summarize the elastic scattering let us note that the scattering potential fixes a boundary condition on the long-range result.
In the next section we will apply this on the initial condition of the problem: the plane wave.

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Scattering Amplitude

The spherical waves are practical to introduce the effect of the scattering into the wavefunction, however, the initial condition is the incoming plane wave ψ(0). In order to use the spherical waves, let us project the plane wave onto the basis of spherical waves. The projection is given by ∞ ∞ ψ(0) = e+ikz = il 4π(2l + 1) jl(kr)Yl 0(θ,φ) = cl ϕl0(r). (1.13) l=0 l=0
The fact that there are only terms with m = 0 in the projection of the plane wave, shows the cylindrical symmetry of the scattering. Here we have made, without loss of generality, the choice of having an incoming plane wave traveling along the z-axis towards the center, which means that θ is defined as the angle between r and z.

Table of contents :

I. Theory 
1. Two-Particle Problem 
1.1. Scattering
1.1.1. Center of Mass (CoM) Motion
1.1.2. Radial Schrödinger Equation
1.1.3. Scattering Potential
1.1.4. Spherical Waves
1.1.5. Elastic Scattering
1.1.6. Scattering Amplitude
1.1.7. Scattering Cross Section
1.1.8. The Unitary Limit
1.1.9. Low-Temperature Limit: Bosons versus Fermions
1.1.10. Scattering Length
1.2. Feshbach Resonances – Tuning the Scattering Length
1.2.1. Two-Channel Model
1.2.2. Determining the Position and Width
1.3. Summary
2. Three-Particle Scattering 
2.1. Elastic Scattering
2.1.1. Three-Particle Hamiltonian
2.1.2. Hyperangular Problem
2.1.3. Scattering Regimes
2.2. Unitary Interactions – Efimov’s Ansatz
2.2.1. Hyperspherical Waves
2.2.2. Short-Distance Scattering – R<Rm Elastic Scattering Efimov Bound States Zero-Range Model
2.2.3. Long-Distance Scattering Long-Range Wavefunction Coupling of the Long-Range to the Short-Range
2.3. Finite-a – Hyperspherical Channels
2.3.1. Long-Distance Scattering Long-Range Wavefunction Coupling of the Long-Range to the Short-Range Effective Two-Channel System
2.4. Inelastic Three-Particle Processus
2.4.1. Elastic versus Inelastic Scattering
2.4.2. Short-Range Elastic → Inelastic Scattering Inelastic Zero-Range Model (ZRM)
2.4.3. Long-Range Resonant Interactions: Efimov Physics Finite Interactions
2.4.4. Flux and Recombination
2.4.5. Temperature Average
2.4.6. Optical Resonator Analogy
2.4.7. Oscillations of L3(T) at Unitarity
2.4.8. Numerical Analysis of L3(T, a)
2.5. Three-Particle Losses on the Positive-a Side
2.5.1. Weakly Bound Dimer
2.5.2. Weakly Bound Dimers and the Efimov Channel
2.5.3. Atom-Dimer Decay with Chemical Equilibrium
2.6. Summary
3. Three-Particle Recombination in a Harmonic Trap 
3.1. Three-Particle Losses in a Trap
3.1.1. Trapping Potential
3.1.2. Weakly and Deeply Bound Dimers in a Trap
3.1.3. Number Decay
3.2. Heating Effects
3.2.1. Weakly-Interaction Limit
3.2.2. Extending the Model to Include Strong Interactions
3.3. Evaporation
3.3.1. A Simple Evaporation Model
3.3.2. More Advanced Model of Evaporation Effects
3.3.3. “Magic” η
3.4. Summary
II. Experiments 
4. The Road to Strongly Interacting Bose Gases 
4.1. Experimental sequence
4.1.1. Lithium-7
4.1.2. Laser System
4.1.3. Zeeman slower
4.1.4. Magneto-Optical Trap (MOT)
4.1.5. Optical Pumping
4.1.6. Magnetic Trapping and Evaporation
4.1.7. Optical Dipole Trap (ODT)
4.1.8. Radio-Frequency (RF) Transitions
4.1.9. Imaging
4.2. Feshbach Resonance in 7Li
4.3. Summary
5. Lifetime of the Resonant Bose Gas 
5.1. Recombination Rate Measurements and Assumptions
5.1.1. Quasi-Thermal Equilibrium Condition
5.1.2. Separation of Time Scales
5.1.3. Starting Point for the Measurements
5.1.4. Number Calibration Pressure calibration Recombination and Temperature calibration
5.1.5. Constant Temperature
5.1.6. Data Analysis
5.2. Results – Unitary Interactions
5.2.1. Temperature Dependence of L3 at Unitarity
5.2.2. Reanalysis using the Advanced Evaporation Model
5.3. Results – Finite Interactions
5.3.1. Saturation of L3 for Resonant Interactions
5.3.2. Comparison with Previous Data – 133Cs The First Efimov Resonance Resonance Position
5.3.3. Temperature Behavior of L3 – 39K Validating the 1/T 2 Law for L3(T) Excess Heat Measurements
5.4. Summary
Concluding remarks 
Appendix A. Technical Details – Theory
A.1. Jacobian and Hyperspherical Coordinates
A.1.1. Jacobian Coordinates
A.1.2. Jacobian → Hyperspherical Coordinates
A.1.3. Jacobian → Hyperspherical Hamiltonian
A.1.4. Hyperradial and Hyperangular Schrödinger Equations
A.2. Incoming and Outgoing waves
A.3. Saddle Point Method
A.4. Efimov’s Ansatz
A.5. The s-matrix at Unitarity
Appendix B. Peer-reviewed papers
B.1. Dynamics and Thermodynamics of the Low-Temperature Strongly
Interacting Bose Gas
B.2. Lifetime of the Bose Gas with Resonant Interactions
B.3. -enhanced sub-Doppler cooling of lithium atoms in D1 gray molasses
Appendix C. Data Loss Measurements
C.1. Unitary Interactions
C.2. Finite-a Interactions
Appendix D. Efimov resonances
D.1. Caesium-133
D.1.1. Universality of the Efimov resonances
D.2. Lithium-7
D.2.1. L3 vs. a
D.3. Rubidium-85
D.4. Potassium-39
D.4.1. Efimov resonance
D.4.2. Universality of the Efimov resonances in 39K
D.5. How to determine the Efimov parameters
D.6. Summary


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